Mouse, Human, Chimpanzee

Transcription

1 More Alignments 1

2 Mouse, Human, Chimpanzee Mouse to Human Chimpanzee to Human 2

3 Mouse v.s. Human Chromosome X of Mouse to Human 3

4 Local Alignment Given: two sequences S and T Find: substrings of S and T that maximizes the alignment score. AATTAG-CCGATGAC TGGAGGCTGATATA I.e., The indels at the beginning and end of the two strings are free. Question: Is the optimal local alignment a local part of an optimal global alignment? 4

5 Warm-up: Prefix alignment We want to find the highest-scoring alignment between two prefixes of the two sequences. AG-CCGATAATT AGGCTGATTGG 5

6 Warm-up: suffix alignment Suppose we only get the free gap at the prefixes of the alignment. AATTAG-CCGAT TGGAGGCTGAT That is, we choose two suffixes, and align them together optimally. Let D[i,j] denote the optimal suffix alignment alignment score of s[1..i], t[1..j]. 6

7 Last column Consider the last column of this optimal suffix alignment. Four cases arise: Case 1: s[i] v.s. t[j] Case 2: s[i] v.s. Case 3: t[j] v.s. Case 4: an empty alignment Case 4 is the only new case comparing to the basic alignment. 7

8 DP algorithm for suffix alignment D[i,j] = max D[i-1, j-1] + f(s[i], t[j]); D[i-1, j] + f(s[i],-); D[i, j-1] + f(-, s[j]); Answer will be here How to backtrace? 8

9 Local Alignment The optimal local alignment is max i,j D[i,j] Suppose the optimal local alignment aligns S[i..i] and T[j..j] together, then it is the best suffix alignment of S[1..i] and T[1..j]. Moreover, any local alignment is a suffix alignment of two prefixes, and vice versa. DP algorithm as before, but backtrack from the maximum element in the matrix. When extended to affined gap, this is the classical Smith-Waterman algorithm. 9

10 Local Alignment 10

11 A Little History The algorithm was first proposed by Temple Smith and Michael Waterman in The global alignment algorithm was called the Needleman-Wunsch algorithm, which was published in

12 An important side note It s fairly straightforward to not just find one local alignment of S and T, but many of them. How? Time complexity? This is crucially important when several sub-regions of S and T are evolutionarily conserved. 12

13 Fit Alignment Given sequence S and T. Find a global alignment between S and a substring of T, maximizing the alignment score. T S Deleting the prefix of T is free, deleting the suffix of T is free. How? Time Complexity? 13

14 Why linear space? Linear Space Alignment Computer RAM used to be very expensive in 80s. Prediction: The cost for 128 kilobytes of memory will fall below U$100 in the near future. Creative Computing magazine. December 1981, page 6. Even today, keeping everything in the L2 cache or within one page of the RAM will speed up the computation. We have learned the linear space if only alignment score, instead of the alignment, is required. We will learn how to get slightly more than the alignment score 14

15 Linear Space Alignment We want to compute one more piece of information the j such that S[1..m/2] aligns with T[1..j] in the optimal alignment. S m/2 T j This is only slightly more than the alignment score. Can this be done in linear space? So what? 15

16 Divide and Conquer S T m/2 j Essentially, j is such that score(s[1..m/2],t[1..j]) + score(s[m/2+1..m],t[j+1,n]) is maximized. We can reverse S and T so that the second part of the formula is also an alignment of prefixes. 16

17 Divide C A T A T T G A T C A T C A A T T G A score(s[1..m/2],t[1.j]) + score(s[m/2+1..m],t[j+1,n]) 17

18 Algorithm Use the previous idea to compute j such that the optimal alignment can be divided into two parts: S[1..m/2] v.s. T[1..j] S[m/2+1..m] v.s. T[j+1..n] Then we use the same algorithm to recursively compute the optimal alignments of these two parts. Return the concatenation of these two optimal alignments. 18

19 Time Complexity T(m,n) mn + T(m/2,j)+T(m/2,n-j) 2 mn 19

20 D.S. Hirschberg, A linear space algorithm for computing longest common subsequences, Comm. ACM 18 (1975)

21 Linear Space Affine Gap Penalty The goal of this paper is to give Hirschberg s idea the visibility it deserves by developing a linear-space version of Gotoh s algorithm. 21

22 Question How to do local alignment in linear space? How to do affined gap penalty in linear space? 22